Set Theory By Naylor and Sell Homework Problems

[fabbi007]

Homework Statement

I am trying understand the approach to the problems given in the following textbook. Linear Operator Theory in Engineering and Science (Applied Mathematical Sciences) (Volume 0) [Paperback]
Arch W. Naylor (Author), George R. Sell (Author). Pages 12-29. There are exercise problems for which I need approach. Please let me know if anyone has a good understanding of them. Appreciate your response.

Homework Equations

The Attempt at a Solution

 

[Mark44]

Speaking for myself, I don't have this book. If you have questions about one of the problems, post it here, along with your attempt at a solution.

 

[fabbi007]

Is Card (N even)< Card (N)? Where N even is set of all even Natural numbers, N is set of all Natural numbers.

Hint: use the mapping from
N even to N is given by n-->n
a. Show examples of this mapping from
N even to N.

b. Is the mapping above onto? One-to-one?

My try at this question:

Since the mapping is n-->n it is the range of this mapping function is a proper subset of N hence the card (N even) <card (N) because there is one to one mapping and the N even maps into N.

Examples above mapping. 2->2, 4->4, 8->8, 268->268... Hence card(N even)< Card

 

[Mark44]

How about this mapping:
2 --> 1
4 --> 2
6 --> 3
8 --> 4
.
.
.
2k --> k
This mapping is 1-to-1 and onto. What does that say about the cardinality of the two sets?

 

[fabbi007]

How about this mapping:
2 --> 1
4 --> 2
6 --> 3
8 --> 4
.
.
.
2k --> k
This mapping is 1-to-1 and onto. What does that say about the cardinality of the two sets?

They are equal!

 

[Mark44]

Yes.